The move will probably mean Brits will have to carry another ID in addition to a passport

Amber Rudd said she was “aware that there is an expectation and concern” about EU citizens already in the UK.

She said: “There will be a need to have some sort of documentation but we are not going to set it out yet.

“We are going to do it in a phased approach to ensure that we use all the technology advantages that we are increasingly able to harness to ensure that all immigration is carefully handled.”

She separately fuelled yet more speculation that the Government is laying the groundwork for a ‘soft’ Brexit by saying she was aware parts of Britain’s economy were dependent on low-skilled migrants – such as farming and agriculture.

She said: “We are aware how necessary it will be to have a seasonal scheme in place and we are looking carefully at it.”

The clamour has sparked a huge backlog in the Government department given that it typically only processes 25,500 permanent residency applications every year.

The Home Office has yet to spell out who may have to register, but it is thought Irish citizens would be exempt.

Officials at the Office for National Statistics earlier this summer admitted they had no idea exactly how many EU nationals may be living in Britain – given that under ‘free movement’ rules they don’t have to register when they arrive in the country.